Previous no-confidence motions against Zuma have failed as the ANC has a commanding majority, but there is a growing backlash against him from within his own party and its allies after a string of missteps which have culminated with his firing of a respected finance minister that rocked markets.
Speaker Baleka Mbete, who is also the national chairperson of the ruling African National Congress (ANC), said in a televised news conference after she cut short a trip abroad that she wanted to ensure the opposition requests were "given the appropriate consideration."
"I must stress that I am alive to the extreme challenges and sense of anxiety our young democracy is going through at this moment," Mbete said.
Analysts say the sacking of finance minister Pravin Gordhan last week may trigger damaging credit downgrades from ratings agencies, which would drive up the government's borrowing costs and restrict its ability to carry out the populist redistribution policies Zuma has indicated he wants to pursue.
Zuma's authority has been steadily undermined as his administration has staggered through a series of blunders and scandals, including the use of state funds to make lavish improvements to his rural homestead and a debacle that threatened the payment of social grants to millions who depend on them.
The ANC, which has governed since the end of apartheid in 1994, is now riven by splits.
Deputy President Cyril Ramaphosa said he disagreed with the decision to sack Gordhan and other top ANC officials have also expressed their dismay at the move.
The South African Communist Party, a key political ally of the ANC, called on Friday for Zuma to resign.
Opposition parties and ANC sources allege that Gordhan was ousted because he was obstructing access to state funds that Zuma and his allies coveted.
Ramaphosa said Zuma removed Gordhan on the basis of a "spurious" intelligence report that accused him and his deputy Mcebisi Jonas, who was also dismissed, of plotting with banks in London to undermine the South African economy. Gordhan dismissed the report as "absolute nonsense."
"We are fast running out of time to save our country from the clutches of the Zuma administration," Mmusi Maimane, leader of the biggest opposition party, the centrist Democratic Alliance (DA), said on Sunday.
The DA and the ultra-left Economic Freedom Fighters Party (EFF) both wrote Mbete asking that she reconvene parliament and hold an urgent sitting for a no-confidence motion.
The ANC has 249 seats in the 400-seat parliament. DA sources say they cannot rely on all opposition members to support the motion, so they probably need 60 MPs from the ruling party to side with them to get the majority needed to topple Zuma.
Zuma, who turns 75 this month, is a traditional Zulu and ex-intelligence operative with an earthy demeanour who has proven himself a political survivor over the years in the face of numerous challenges, including hundreds of corruption charges that were dropped in 2009 but could still be reinstated.
(Reporting by Ed Stoddard, Editing by Jeremy Gaunt and Greg Mahlich)
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)